This paper is concerned with algorithmic techniques for the incremental generation of continuous curves that can efficiently cover an abstract surface. We introduce the notion of low-discrepancy curves as an extension of the notion of low-discrepancy sequences such that sufficiently smooth curves with low-discrepancy properties can be defined and generated. We then devise a procedure for lifting these curves, that efficiently cover the unit cube, to abstract surfaces, such as nonlinear manifolds. We present algorithms that yield suitable fair mappings between the unit cube and the abstract surface. We demonstrate the application of these ideas using some concrete examples of interest in robotics.